Hi Kayleigh, I was reading on Wikipedia for info regarding your question and it seems that begonia societies have come up with these terms and that there often is crossover of characteristics, like leaf size, shape, textures, wiki doesn't clearly state what a rex begonia is.
In my mind I think of them as having larger leaves and less dramatic flowers. Wax begonias are one of the different types; rex, wax, hardy, cane stem, rhizomatous.
Hope that helps